Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $10^{4}\cdot40^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40A8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.8.938 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&4\\31&27\end{bmatrix}$, $\begin{bmatrix}3&24\\33&21\end{bmatrix}$, $\begin{bmatrix}11&4\\9&19\end{bmatrix}$, $\begin{bmatrix}27&20\\30&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.cn.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{18}\cdot5^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}$ |
| Newforms: | 50.2.a.a, 50.2.a.b$^{3}$, 200.2.a.a, 200.2.a.e, 400.2.a.d, 400.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ 2 x t - x u + y u - w v + w r $ |
| $=$ | $x t + 2 y u - z w - z v$ | |
| $=$ | $x w + x r + y w + y v - 2 w u$ | |
| $=$ | $2 x^{2} + x y + y^{2} - z w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 256 x^{10} + 288 x^{8} y^{2} + 1440 x^{8} z^{2} - 65 x^{6} y^{4} - 1690 x^{6} y^{2} z^{2} + \cdots + 500 y^{4} z^{6} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:-1:-1:1)$, $(0:0:0:0:0:1:-1:1)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 20.60.4.l.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w-v$ |
| $\displaystyle W$ | $=$ | $\displaystyle w+r$ |
Equation of the image curve:
| $0$ | $=$ | $ 35X^{2}+5Y^{2}-Z^{2}+W^{2} $ |
| $=$ | $ 5X^{3}-5XY^{2}+XZ^{2}-YZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.120.8.cn.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}t$ |
Equation of the image curve:
| $0$ | $=$ | $ -256X^{10}+288X^{8}Y^{2}+1440X^{8}Z^{2}-65X^{6}Y^{4}-1690X^{6}Y^{2}Z^{2}-5225X^{6}Z^{4}+35X^{4}Y^{6}+245X^{4}Y^{4}Z^{2}+2150X^{4}Y^{2}Z^{4}+9000X^{4}Z^{6}-3X^{2}Y^{8}-20X^{2}Y^{6}Z^{2}-325X^{2}Y^{4}Z^{4}-3000X^{2}Y^{2}Z^{6}-10000X^{2}Z^{8}+Y^{10}+25Y^{8}Z^{2}+200Y^{6}Z^{4}+500Y^{4}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-40.bn.1.10 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-20.l.1.2 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-20.l.1.4 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.bl.1.6 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bl.1.12 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bn.1.10 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bn.1.16 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.480.16-40.cc.1.4 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.cc.2.2 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.cd.1.4 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.480.16-40.cd.2.2 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.720.22-40.fl.1.12 | $40$ | $3$ | $3$ | $22$ | $2$ | $1^{14}$ |
| 40.960.29-40.xf.1.10 | $40$ | $4$ | $4$ | $29$ | $4$ | $1^{21}$ |
| 80.480.16-80.by.1.14 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.by.2.8 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.bz.1.12 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.bz.2.8 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.17-80.bk.1.1 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.bm.1.5 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.dq.1.7 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.ds.1.3 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.16-120.fs.1.8 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.fs.2.12 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.ft.1.8 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.ft.2.8 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.co.1.22 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.co.2.14 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.cp.1.22 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.cp.2.14 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.17-240.dy.1.1 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.ea.1.5 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.ju.1.18 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.jw.1.22 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.16-280.eo.1.16 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.eo.2.12 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.ep.1.16 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.ep.2.8 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |